Using Pearls shorthand \(\hat{x} = \textrm {do}(X=x)\)  for an intervention, we can easily predict the effect of an intervention on \(y\)  from their joint distribution \(P(X,Y)\) . For instance, \(P(y | \hat{x}, h_{X \rightarrow Y}) = P(p |x)\)  [1]}. Therefore, we can compute the posterior probability of \(h\)  given an intervention \(\hat{x}\)  resulted in observing \(Y=y\) :
\( P (h|\hat{x},y)&= \frac{ P (y|h,\hat{x}) P (\hat{x}|h) P (h)}{ \sum _{h \in \lbrace  h_{X \rightarrow Y}, h_{Y\rightarrow X}\rbrace  }  P (y|h,\hat{x}) P (\hat{x}|h) P (h) }\nonumber \\& = \frac{P(y | x)P(h) }{ P(y|x)P(h) + P(y ) P(\lnot h) } \) 
